All that remains is bounding the norm involving the noise terms with high probability, which is the second term in the result stated in Lemma REF . By Fenchel Duality, we have that [1]}:
\(\frac{1}{2}{S_{t}}_{\bar{W}_{t}^{-1}}^{2} &= \max _{x\in \mathbb {R}^{d}}\left\lbrace \langle x, S_{t}\rangle - \frac{1}{2}{x}_{\bar{W}_{t}}^{2}\right\rbrace \nonumber \\&= \max _{x\in \mathbb {R}^{d}}\left\lbrace \log \left(\exp \left(\langle x, S_{t}\rangle - \frac{1}{2}{x}_{\bar{W}_{t}}^{2}\right)\right)\right\rbrace \nonumber \\&= \log \left(\max _{x\in \mathbb {R}^{d}}\left\lbrace \exp \left(\langle x, S_{t}\rangle - \frac{1}{2}{x}_{\bar{W}_{t}}^{2}\right)\right\rbrace \right)\) 
